# background and BNM matrix model.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Multiple M
-
wave in 11D pp
-
wave
background and BNM matrix model.

Igor A. Bandos

Based on paper in preparation (2011), Phys.Lett.
B687

(2010),
Phys. Rev. Lett.

105
(2010), Phys, Rev
. D82
(2010).

IKERBASQUE, The Basque Foundation for Science

and

Depto de Física Teórica, Universidad del País Vasco UPV/EHU
, Bilbao, Spain

-

Introduction

-

Superembedding approach to M0
-
brane (M
-
wave) and multiple M0
-
brane system (
mM0
= multiple M
-
wave).

-

Equations of motion for mM0 in generic 11D SUGRA background

-

Equations of motion for mM0 in supersymmetric 11D pp
-
wave
background and
Berenstein
-
Maldacena
-
Nastase (
BMN
)
Matrix model.

-

Conclusion and outlook.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Introduction

Matrix model was proposed by
Banks, Fischler, Shenker and Susskind

in
96
-

and remains an important tool for studying M
-
theory.

Also the theory is 11 dimensional, original BFSS Lagrangian is just a
dimensional reduction of D=10 SYM down to d=1(low energy mD0). The
symmetry enlargement to D=11 Lorentz symmetry was reveiled by BFSS.

However it was not clear how to write the action for Matrix model in 11D
supergravity background.

This is why matrix models are known (were guessed) for a few particular
supergravity background, in particular

for pp
-
wave background [
Berenstein
-
Maldacena
-
Nastase 2002
] =BMN
Matrix model

for matrix Big Bang background [Craps, Sethi, Verlinde 2005]

A natural way to resolve this problem was to obtain invariant action, or
covariant equations of motion, for multiple M0
-
brane system. But it was a
problem to write such action.

Purely bosonic mM0 action [
Janssen & Y. Losano

2002]

similar to
Myers ‘dielectric’ ‘mDp
-
brane’ , neither susy nor Lorentz invariance.

Superembedding approach to
mM0

system

[
I.B
.

2009
-
2010] (multiple M
-
waves or multiple massless superparticle in 11D) =>
SUSY inv. Matrix
model equations in an arbitrary 11D SUGRA.

This talk begins the program of develop the application of this result.

It is natural to begin by specifying the general equations for pp
-
wave
background and compear with equations of the BMN matrix model.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Matrix model eqs. in general 11D
SUGRA background
[I.B. 2010]

Obtained in the frame of superembedding approach

So we need to say what is the superembedding approach

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Bosonic particles and p
-
branes
.

W

M

Target
space =

spacetime

Worldvolume (worldline)

These worldvolume fields carrying vector indices of D
-
dimensional spacetime

coordinates are restricted by the
p
-
brane equations of motion

The embedding of
W

in
Σ

can be

described by coordinate functions

minimal surface equation

(Geodesic eq. for p=0)

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Super
-
p
-
branes. Target superspaces and worldvolume

W

Σ

Target D(=11) SUPERspace

Worldvolume (worldline)

The embedding of
W

in
Σ

can be
described by coordinate
functions

These bosonic and fermionic worldvolume fields, carrying indices of 11D superspace

coordinates, are restricted by the super
-
p
-
brane equations of motion

Grassmann or

fermionic coordinates

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Superembedding approach,

[Bandos,Pasti.Sorokin, Tonin, Volkov 1995, Howe, Sezgin 1996],

following the pioneer STV approach to superparticle and
superstring
[Sorokin, Tkach, Volkov MPLA 1989],

provides a superfield description of the super
-
p
-
brane
dynamics, in terms of embedding of superspaces,

namely of embedding of worldvolume superspace

into a target superspace

It is thus doubly supersymmetric, and the worldline (world
-
volume) susy repaces
[STV 89]

the enigmatic local
fermionic kappa
-
symmetry
[de Azcarraga+Lukierski 82, Siegel
93]

of the standard superparticle and super
-
p
-
brane action.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

(M)p
-
branes in superembedding approach.

I. Target and worldline superspaces.

W

Σ

Target D=11 SUPERspace

Worldvolume (worldline)
SUPERspace

The embedding of
W

in
Σ

can be
described by coordinate functions

These worldvolume superfields carrying indices of 11D superspace coordinates

are restricted by the
superembedding equation
.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

M0
-
brane (
M
-
wave
) in superembedding approach.

I. Superembedding equation

W

Σ

Supervielbein of D=11
superspace

Superembedding equation

states that the pull
-
back of bosonic vielbein

has vanishing fermionic projection:
.

Supervielbein of worldline
superspace

General decomposition of the pull
-
back of 11D supervielbein

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Superembedding equation

(+ conv. constr.) geometry of worldline superspace

SO(1,1) curvature of

vanishes,

4
-
form flux of 11D SG= field strength of 3
-
form

Moving frame vectors

The 4
-
form flux superfield enters the solution of the 11D superspace SUGRA constraints
[Cremmer & Ferrara 80, Brink & Howe 80] (which results in SUGRA eqs. of motion):

Equations of motion of (single) M0
-
brane

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Moving frame and spinor moving frame variables

appear in the conventional constraints determining the induced supervielbein so that

Equivalent form of the superembedding eq
.

+ conventional constraints.

M0 equations of motion

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Multiple M0 description by d=1,

=16
SU(N) SYM on

.

We describe the multiple M0 by 1d

=16 SYM on

The embedding of into the 11D SUGRA superspace
is determined by the superembedding equation

‘center of energy’ motion of the mM0 system is defined by the single M0 eqs

,

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Multiple M0 description by d=1,

=16 SYM on

Basic SYM constraints and superembedding
-
like equation.

is an su(N) valued 1
-
form potential on

with the field strength

We impose constraint

A clear candidate for the description of relative

motion of the mM0
-
constituents!

Bianchi identities DG=0 the
superembedding
-
like equation

Studying its selfconsistecy conditions, we find the dynamical equations describing

the relative motion of mM0 constituents.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Equations for the relative motion of multiple M0

in an
arbitrary
supergravity background

follow from the constr.

1d Dirac equation

Gauss constraint

Bosonic equations of motion

Coupling to higher form characteristic for the

Emparan
-
Myers dielectric brane effect

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

mM0 in pp
-
wave background

Supersymmeric bosonic pp
-
wave solution of 11D SUGRA

is very well known:

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

mM0 in pp
-
wave background

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Thus it looks like we just have to substitute definite pure
bosonic expressions into the general mM0 equations.

However, this is not the case, because, for instance

and to find some details on the worldline SSP embeddded in

and also (in a more general case of, e.g., non
-
constant flux)

as this allows to find

To specify our mM0 eqs for some particular SUGRA background
it is necessary to describe this background as a superspace

Thus it is not sufficient to know pure bosonic supersymmetric solution

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Fortunately the pp
-
wave superspace is a coset, so that

one can write a definite expression for supervielbein etc. (but…)

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

The worldline (center of energy) superspace is embedded in
this SSP.

Its embedding is specified in terms of bosonic and fermionic
coordinate superfields

Part of these are Goldstone (super)fields corresponding to
(super)symmetries broken by brane/by center of energy of mM0

and part can be identified with coordinates of

for instance,

Goldstone fermion superfield

Let us begin by the simplest case when the Goldstone
(super)fields describing the center of energy motion are =0, i.e.
by describing a vacuum solution for center of energy SSP

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

The embedding of the vacuum worldline superspace

is characterized by that all Goldstone fields are zero or const.

by constant moving frame and spinor moving frame variables

by

more precisely:

One can check that equations of motion and superembedding
equation are satisfied,

With which the flux pull
-
back to
W
is:

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

But our main interest is in intrinsic geometry of

as the relative motion is described on this superspace.

Furthermore, the induced SO(9) and SO(1,1) connection have
only fermionic components, so that

What is really important:

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Now we are ready to specify the Matrix
model equations in general 11D SUGRA SSP

1d Dirac equation

Gauss constraint

Bosonic equations of motion

for the case of completely SUSY pp
-
wave background

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

mM0 eqs in pp
-
wave background

These eqs coincide with the ones which can be obtained by varying the

BMN action up to the fact that they are formulated for traceless matrices.

The trace part of the matrices should describe the center of energy motion.

In our approach it is described separately by the geometry of

To find this, one should go beyond the ground state solution of the superembedding eq,

which we have used above. This is under study now.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Conclusions and outlook

After reviewing of the superembedding approach to mM0
system and the generalization of Matrix model eqs. in an
arbitrary 11D SUGRA background obtained from it

we used them to obtain the mM0 equations of motion in the
supersymmetric pp
-
wave background.

The final answer is obtained for a particular susy solution of
the center of energy equations of motion.

The equations of the relative motion of mM0 constituents
coincide with the BMN equations, but written for traceless
matrices.

To compare the complete set of equations, including the trace
part of the matrices which describe relative motion of the mM0
constituents we need to find general solution of the
superembedding approach equations to M0
-
brane.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Some other directions for future study

Matrix model equations in AdS(4)xS(7) and AdS(7)xS(4), and their
application, in particular in the frame of AdS/CFT.

Thanks for your attention!

Extension of the approach for higher p mDp
-

and mMp
-

systems
(mM2
-
?, mM5
-
?). Is it consistent to use the same construction (SU(N)
SYM on w/v superspace of a single brane)? And, if not, what is the
critical value of p?

To compare the complete set of equations, including the trace
part of the matrices which describe relatove motion of the mM0
constituents we need to find general solution of the
superembedding approach equations to M0
-
brane.

This problem is under investigation now.

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Thank you for

your attention!

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Appendix A: On BPS equations

for the supersymmetric pure bosonic solutions of mM0 equations

½ BPS equation (16 susy’s preserved)

SUSY preservation by

center of energy motion

SUSY preservation by relative

motion of mM0 constituents

has fuzzy S
² solution modeling M2 brane by mM0 configuration

1/4 BPS equation (8 susy’s) with SO(3) symmetry

and

Nahm equation

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

¼ BPS equation (8 susy’s preserved)

which has a(nother) fuzzy 2
-
sphere
-
related solution

The famous Nahm equation

appears as an SO(3) inv

with

and

with

is obeyed, in particular, for

½ BPS:

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Spinor moving frame superfields, entering

are elements of the Spin group valued matrix

`covering’ the moving frame matrix as an SO(1.10) group element

This is to say they are `square roots’ of the light
-
like moving frame variables, e.g.:

One might wander

whether these spinor moving frame variables come from?

(Auxiliary) moving frame superfields

are elements of the Lorentz group valued matrix
.

This is to say they obey

,

Appendix B: Moving frame and spinor moving frame

Miami 2011

I. Bandos, mM0 in pp
-
wave SSP

Supersymmetric extended objects
-

super
-
p
-
branes
-

and multiple brane
systems play important role in String/M
-
theory, AdS/CFT etc.

Single p
-
brane actions

are known for years (84
-
97)

Multiple p
-
brane actions

(multiple superparticle action for p=0):

Multiple Dp
-
branes (mDp): (very) low energy limit = U(N) SYM (1995)

In search for a complete (a more complete) supersymmetric,
diffeomoprhism and Lorenz invariant action =only a particular progress:

-
purely bosonic mD9 [
Tseytlin
]: non
-
Abelian BI with symm. trace (no susy);

-

Myers

 a
`dielectric brane action

= no susy, no Loreentz invartiance(!)

-

Howe, Lindsrom and Wulff

[2005
-
2007]: the
boundary fermion approach
.
Lorentz and susy inv. action for mDp,
but

on the ‘minus one quantization level’.

-

I.B.

2009
-

superembedding approach to

mD0
(proposed for mDp, done for p=0)

Multiple Mp
-
branes (mMp): even more complicated.

-

purely bosonic mM0= [
Janssen & Y. Losano

2002])

-

(very) low energy limit = BLG (2007; 3
-
algebras) ABJM (2008)

-

nonlinear generalization of (Lorentzian) BLG =[
Iengo & J. Russo

2008]

-

low energy limit of mM5 = mysterious (2,0) susy d=6 CFT ????

-

Superembedding approach to
mM0

system

[
I.B
.

2009
-
2010] (multiple M
-
waves or multiple massless superparticle in 11D) =>
SUSY inv Matrix model
equations in an arbitrary 11D SUGRA

(
this talk develops its application
).

Introduction